Group #24906259 — subgroups

Subgroups — C₈×D₆₆₄ #24906259

Order 10624.

196 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₈×D₃₃₂	C₂	1	2	yes	maximal, complemented
3	#19127454	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127430	C₂	1	2	yes	maximal, complemented
5	C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₆₄	C₄	1	1	yes	characteristic
7	C₂×C₈×D₈₃	—	2	2	no	—
8	C₄×D₃₃₂	C₂²	1	2	yes	—
9	#14581726	C₄	1	1	yes	characteristic
10	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
11	#14581714	C₄	1	2	yes	—
12	(C₂×D₈₃)⋊C₈	—	2	2	no	—
13	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₈₃	—	2	2	no	—
18	D₆₆₄	C₈	1	4	yes	direct product factor
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₈×D₈₃	—	4	4	no	—
21	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
22	(C₂×D₈₃)⋊C₄	—	2	2	no	—
23	C₂×C₈×C₈₃	—	2	1	no	—
24	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₄²×C₈₃	C₂³	1	1	yes	characteristic
29	C₂²×D₈₃	—	2	2	no	—
30	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
33	C₂×Dic₈₃	—	2	2	no	—
34	D₃₃₂	—	2	2	no	—
35	D₃₃₂	C₂×C₈	1	4	yes	complemented
36	C₄×D₈₃	—	4	2	no	—
37	C₄×D₈₃	—	4	2	no	—
38	C₈×C₈₃	—	2	2	no	—
39	C₈×C₈₃	—	2	1	no	—
40	C₈×C₈₃	—	2	1	no	—
41	C₈×C₈₃	C₂×C₈	1	2	yes	complemented
42	C₈×C₈₃	D₈	1	2	yes	complemented
43	C₈₃⋊C₈	—	4	2	no	—
44	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
45	C₂×D₈₃	—	2	4	no	—
46	C₂×D₈₃	—	4	2	no	—
47	C₂×D₈₃	—	4	2	no	—
48	C₄×C₈₃	—	2	1	no	—
49	C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₈₃	#131	1	1	yes	characteristic
51	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₈₃	#135	1	1	yes	characteristic
53	Dic₈₃	—	4	2	no	—
54	C₂×C₈₃	C₈×D₄	1	1	yes	characteristic
55	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
56	C₂×C₈₃	#443	1	1	yes	characteristic
57	D₈₃	—	4	4	no	—
58	C₈×D₈	—	83	1	no	maximal, Sylow
59	C₈₃	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
60	C₄×D₈	—	83	1	no	—
61	C₈×D₄	—	83	2	no	—
62	#335	—	83	1	no	—
63	D₄⋊C₈	—	83	2	no	—
64	C₈²	D₈₃	1	1	yes	characteristic, complemented
65	C₂×D₈	—	83	1	no	—
66	C₂²×C₈	—	166	2	no	—
67	C₄×D₄	—	83	2	no	—
68	#107	—	83	1	no	—
69	C₄⋊C₈	—	83	2	no	—
70	#102	—	83	2	no	—
71	C₂²⋊C₈	—	166	2	no	—
72	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
73	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
74	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
75	C₂×D₄	—	83	2	no	—
76	C₂²×C₄	—	166	2	no	—
77	D₈	—	83	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	332	4	no	—
80	C₂×C₈	—	166	2	no	—
81	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
82	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
83	C₂×C₈	D₃₃₂	1	1	yes	characteristic
84	C₂×C₈	D₃₃₂	1	1	yes	characteristic
85	C₄⋊C₄	—	83	2	no	—
86	C₂²⋊C₄	—	166	2	no	—
87	C₄²	C₂²×D₈₃	1	1	yes	characteristic
88	C₂³	—	166	2	no	—
89	D₄	—	83	4	no	—
90	D₄	—	166	2	no	—
91	C₂×C₄	—	332	2	no	—
92	C₂×C₄	—	332	2	no	—
93	C₂×C₄	—	166	2	no	—
94	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
95	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
96	C₂×C₄	#13050590	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	332	2	no	—
101	C₈	C₈×D₈₃	1	2	yes	complemented
102	C₈	D₆₆₄	1	2	yes	direct product factor
103	C₂²	—	332	2	no	—
104	C₂²	—	332	2	no	—
105	C₂²	—	166	4	no	—
106	C₂²	C₄×D₃₃₂	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	332	2	no	—
109	C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
110	C₄	#14581798	1	1	yes	characteristic
111	C₄	C₂×D₆₆₄	1	1	yes	characteristic
112	C₄	#14581814	1	1	yes	characteristic
113	C₂	—	332	4	no	—
114	C₂	C₈×D₃₃₂	1	1	yes	characteristic
115	C₂	C₄×D₆₆₄	1	1	yes	characteristic
116	C₂	#19127819	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₆₆₄

C₂

yes

maximal, characteristic

C₈×D₃₃₂

C₂

yes

maximal, complemented

#19127454

C₂

yes

maximal, characteristic, complemented

#19127430

C₂

yes

maximal, complemented

C₈²×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×D₆₆₄

C₄

yes

characteristic

C₂×C₈×D₈₃

—

C₄×D₃₃₂

C₂²

yes

—

#14581726

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581714

C₄

yes

—

(C₂×D₈₃)⋊C₈

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×D₃₃₂

C₂×C₄

yes

—

C₂×C₄×D₈₃

—

D₆₆₄

C₈

yes

direct product factor

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

(C₂×D₈₃)⋊C₄

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

C₂×C₈

yes

complemented

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₈

yes

complemented

C₈×C₈₃

D₈

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²×C₈

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₄×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

Dic₈₃

—

C₂×C₈₃

C₈×D₄

yes

characteristic

C₂×C₈₃

C₄×D₈

yes

characteristic

C₂×C₈₃

#443

yes

characteristic

D₈₃

—

C₈×D₈

—

maximal, Sylow

C₈₃

C₈×D₈

yes

characteristic, complemented, Sylow

C₄×D₈

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

D₈₃

yes

characteristic, complemented

C₂×D₈

—

C₂²×C₈

—

166

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₄×C₈

C₂×D₈₃

yes

characteristic

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

166

—

D₈

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

—

166

—

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

D₃₃₂

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

—

166

—

D₄

—

D₄

—

166

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

#13050590

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

332

—

101

C₈

C₈×D₈₃

yes

complemented

102

C₈

D₆₆₄

yes

direct product factor

103

C₂²

—

332

—

104

C₂²

—

332

—

105

C₂²

—

166

—

106

C₂²

C₄×D₃₃₂

yes

characteristic

107

C₄

—

108

C₄

—

332

—

109

C₄

C₂×C₈×D₈₃

yes

characteristic

110

C₄

#14581798

yes

characteristic

111

C₄

C₂×D₆₆₄

yes

characteristic

112

C₄

#14581814

yes

characteristic

113

C₂

—

332

—

114

C₂

C₈×D₃₃₂

yes

characteristic

115

C₂

C₄×D₆₆₄

yes

characteristic

116

C₂

#19127819

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₃₃₂
commutator	C₄×C₈₃	C₂²×C₈
Frattini	C₄²	C₂²×D₈₃
Fitting	C₈²×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₆₆₄	C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₈²	D₈₃
hypocenter	C₈₃	C₈×D₈
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₈

D₃₃₂

commutator

C₄×C₈₃

C₂²×C₈

Frattini

C₄²

C₂²×D₈₃

Fitting

C₈²×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₆₆₄

C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₈²

D₈₃

hypocenter

C₈₃

C₈×D₈

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. C₈×D₃₃₂	C₂	semidirect
3. #19127454	C₂	semidirect
4. #19127430	C₂	semidirect
5. C₈²×C₈₃	C₂	semidirect
18. D₆₆₄	C₈	direct (N × H)
35. D₃₃₂	C₂×C₈	semidirect
41. C₈×C₈₃	C₂×C₈	semidirect
42. C₈×C₈₃	D₈	semidirect
59. C₈₃	C₈×D₈	semidirect
64. C₈²	D₈₃	semidirect
101. C₈	C₈×D₈₃	semidirect
102. C₈	D₆₆₄	direct (N × H)

H ≅ G/N

type

2. C₈×D₃₃₂

C₂

semidirect

3. #19127454

C₂

semidirect

4. #19127430

C₂

semidirect

5. C₈²×C₈₃

C₂

semidirect

18. D₆₆₄

C₈

direct (N × H)

35. D₃₃₂

C₂×C₈

semidirect

41. C₈×C₈₃

C₂×C₈

semidirect

42. C₈×C₈₃

D₈

semidirect

59. C₈₃

C₈×D₈

semidirect

64. C₈²

D₈₃

semidirect

101. C₈

C₈×D₈₃

semidirect

102. C₈

D₆₆₄

direct (N × H)

Subgroups — C₈×D₆₆₄ #24906259

Conjugacy classes of subgroups

Complements

Subgroups — C8×D664 #24906259

Conjugacy classes of subgroups

Complements

Subgroups — C₈×D₆₆₄ #24906259